Дано: ∠B = 60°, AB : BC = 7 : 5. Найти: ∠A, ∠C, ∠AOC.

Решение:

• \[ \cup AB : \cup BC = 7:5 \]
• \[ \angle B = 60^{\circ} \]
• \[ \angle AOC = \frac{1}{2} \cup AC \]
• \[ \cup AB + \cup BC = \text{полный круг} - \cup AC \]
• \[ \angle B = \frac{1}{2} \cup AC \]
• \[ 60^{\circ} = \frac{1}{2} \cup AC \]
• \[ \cup AC = 120^{\circ} \]
• \[ \angle AOC = \frac{1}{2} \cdot 120^{\circ} = 60^{\circ} \]
• \[ \cup AB = 7x, \quad \cup BC = 5x \]
• \[ 7x + 5x = 360^{\circ} - 120^{\circ} \]
• \[ 12x = 240^{\circ} \]
• \[ x = 20^{\circ} \]
• \[ \cup AB = 7 \cdot 20^{\circ} = 140^{\circ} \]
• \[ \cup BC = 5 \cdot 20^{\circ} = 100^{\circ} \]
• \[ \angle A = \frac{1}{2} \cup BC = \frac{1}{2} \cdot 100^{\circ} = 50^{\circ} \]
• \[ \angle C = \frac{1}{2} \cup AB = \frac{1}{2} \cdot 140^{\circ} = 70^{\circ} \]

Ответ: ∠A = 50°, ∠C = 70°, ∠AOC = 60°.